Nature of problem:Planar channeling radiation is emitted by relativistic charged particles during traversing a single crystal in direction parallel to a crystallographic plane. Channeling is modeled as the motion of charged particles in a continuous planar potential which is formed by the spatially and thermally averaged action of the individual electrostatic potentials of the crystal atoms of the corresponding plane. Classically, the motion of channeled particles through the crystal resembles transverse oscillations being the source of radiation emission. For electrons of energy less than 100 MeV considered here, planar channeling has to be treated quantum mechanically by a one-dimensional Schrödinger equation for the transverse motion. Hence, this motion of the channeled electrons is restricted to a number of discrete (bound) channeling states in the planar continuum potential, and the emission of channeling radiation is caused by spontaneous electron transitions between these eigenstates. Due to relativistic and Doppler effects, the energy of the emitted photons directed into a narrow forward cone is typically shifted up by about three to five orders of magnitude. Consequently, the observed energy spectrum of channeling radiation is characterized by a number of radiation lines in the energy domain of hard X-rays. Channeling radiation may, therefore, be applied as an intense, tunable, quasi-monochromatic X-ray source.

Solution method:The problem consists in finding the electron wave function for the planar continuum potential. Both the wave functions and corresponding energies of channeling states solve the Schrödinger equation of transverse electron motion. In the framework of the so-called many-beam formalism, solving the Schrödinger equation reduces to a eigenvector-eigenvalue problem of a Hermitian matrix. For that the program employs the mathematical tools allocated in the commercial computation software Mathematica. The electric field of the atomic planes in the crystal forces dipole oscillations of the channeled charged particles. In the quantum mechanical approach, the dipole approximation is also valid for spontaneous transitions between bound states. The transition strength for dedicated states depends on the magnitude of the corresponding dipole matrix element. The photon energy correlates with the particle energy, and the spectral width of radiation lines is a function of the life times of the channeling states.

Running time:The program has been tested on a PC AMD Athlon X2 245 processor
2.9 GHz with 2 GB RAM. Depending on electron energy and crystal thickness,
the running time of the program amounts to 5-10 minutes.